On the optimality of the Gittins index rule for multi-armed bandits with multiple plays

We investigate the general multi-armed bandit problem with multiple servers. We determine a condition on the reward processes su1⁄2cient to guarantee the optimality of the strategy that operates at each instant of time the projects with the highest Gittins indices. We call this strategy the Gittins index rule for multi-armed bandits with multiple plays, or brie ̄y the Gittins index rule. We show by examples that: (i) the aforementioned su1⁄2cient condition is not necessary for the optimality of the Gittins index rule; and (ii) when the su1⁄2cient condition is relaxed the Gittins index rule is not necessarily optimal. Finally, we present an application of the general results to the multiserver scheduling of parallel queues without arrivals.